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MATH3203 Lecture 3 PDEs in Biology and Classification of PDEs
MATH3203 Lecture 3 PDEs in Biology and Classiﬁcation of PDEs Dion Weatherley Earth Systems Science Computational Centre, University of Queensland March 6, 2006 Abstract Contents 1 Recap of previous lecture 2 1.1 PDE form of the fundamental conservation law . . . . . . . . . . . . . . . 2 1.2 Advection and diﬀusion equations . . . . . . . . . . . . . . . . . . . . . . . 2 2 PDE models in Biology 3 2.1 Reﬁnement of the population model . . . . . . . . . . . . . . . . . . . . . . 3 2.2 Other PDE models in the life sciences . . . . . . . . . . . . . . . . . . . . . 4 3 Types of BCs and PDE classiﬁcation 4 3.1 Types of PDE Boundary Conditions . . . . . . . . . . . . . . . . . . . . . 4 3.1.1 Dirichlet Boundary Conditions . . . . . . . . . . . . . . . . . . . . . 4 3.1.2 Neumann Boundary Conditions . . . . . . . . . . . . . . . . . . . . 5 3.1.3 Radiation Boundary Conditions . . . . . . . . . . . . . . . . . . . . 5 3.2 Classiﬁcation of PDEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 3.3 Characteristic coordinates and canonical forms . . . . . . . . . . . . . . . . 6 3.3.1 Hyperbolic case, D > 0 . . . . . . . . . . . . . . . . . . . . . . . . . 7 3.3.2 Parabolic case, D = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . 7 3.3.3 Elliptic case, D < 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1 1 Recap of previous lecture 1.1 PDE form of the fundamental conservation law Commencing with the principle of conservation of a quantity within a given domain, we derived the PDE formulation of the fundamental conservation law: εt = − .φ + σ. (1) where ε(x, t) represents the density of the conserved quantity for each point in the domain (x ∈ Ω) at any time, t. The ﬂux of the conserved quantity through any given point is given by the ﬂux vector, φ(x, t) and the density of sources or sinks of the conserved quantity is given by σ(x, t). Although we derived the fundamental conservation law assuming the conserved quan- tity was energy, the same form applies for any conserved quantity. It is useful when analysing a particular PDE, to consider it as a conservation law and identify the terms playing the roles of density, ﬂux and sources. 1.2 Advection and diﬀusion equations The fundamental conservation law is a single equation in three unknowns, a so-called underconstrained problem. In order to specify a properly constrained PDE model, we must introduce constitutive relations relating the ﬂux or sources to the density. Constitu- tive relations arise from assumptions about the physical properties of the medium within the domain. We examined two constitutive laws for the ﬂux: advection and diﬀusion constitutive laws. The advection constitutive law assumes that the conserved quantity is carried by an underlying velocity ﬁeld, v(x) that is independant of the density of the conserved quantity. The ﬂux term for advection is given by: φ(x, t) = ε(x, t)v(x) (2) It is common to further assume that the ﬂow is incompressible which is expressed as .v = 0, leading to the incompressible ﬂow advection equation: εt + v. ε=0 (3) By solving the 1D advection equation, we demonstrated that advection models unde- formed transport of the conserved quantity through the domain. The second constitutive law we examined was the diﬀusion equation, modelling the ﬂux of the conserved quantity from regions of high density to regions of lower density. In this case the ﬂux is proportional to the spatial gradient of the density: φ(x, t) = −D ε(x, t) (4) When the constitutive law is substituted into the fundamental conservation law, we obtain the diﬀusion equation: 2 εt − D ε=0 (5) 2 where is the Laplace operator: 2 2 δ2 δ2 δ2 = + 2+ 2 (6) δx2 δy δz Constitutive laws may be combined to model more complex situations. To model both advection and diﬀusion, we can sum the appropriate constitutive laws for the ﬂux, to obtain the advection-diﬀusion equation: 2 εt − D ε+ε .v + v. ε=0 (7) 2 PDE models in Biology In the physical sciences and engineering, mathematical models are usually expressed in terms of nearly exactly measurable quantities and the goal is to obtain precise quanti- tative results to be compared with experimental observations. In the biological sciences however, it is rare that the phenomenon to be studied can be so precisely quantiﬁed, due to the complexity of the systems to be considered. It is diﬃcult to account for all the intracacies of predator-prey interactions, tumor growth or the spread of infectious diseases for example. Typically biological models are more qualitative, aiming to predict only quanitative features rather than yielding detailed quantitative results. This section aims to introduce some standard PDE models in the biological sciences, as a complement to our derivation of advection and diﬀusion equations from physical principles. 2.1 Reﬁnement of the population model In the ﬁrst lecture, we studied the Malthus and Logistic models for population growth. Both such models were ODE models, involving only one independant variable, time. We introduced the physical variable U (t) to represent the size of the population. It is implicitly understood however, that this variable represents the population density (or population per unit volume). While both these models capture some aspects of population growth, both ignore the spatial distribution of an organism within its ecosystem. Even for an ecosystem with only a single species, population pressure will cause the organisms to migrate apart, to minimise the impact of competition for resources. To attempt to model this, we must extend our deﬁnition of the population density to U (x, t), representing the population density within a small volume surrounding x at time t. Implicit in the model is the assumption that there are suﬃcient organisms that deﬁning a local population density is reasonable. To model population changes in both space and time, we must introduce a PDE population growth model that includes terms governing the motion of organisms within the ecosystem. One of the fundamental PDE models in biology is Fisher’s equation. This equation assumes that the population will spread out spatially within a 1D domain. Fisher’s equation is: U Ut = DUxx + rU 1 − , U = U (x, t) (8) K By comparison with the fundamental conservation law, we can identify this as a diﬀusion equation, with an additional source term modelling population growth due to reproduc- tion. 3 2.2 Other PDE models in the life sciences In the biological sciences, it is not always necessary to model the spatial structure however one might be interested in modelling situations with other types of structure. For example, we might be interested in modelling the ages of the organisms. Then the population density will depend on both the age (a) and the time (t) i.e. U = U (a, t). In life history theory, models are developed that predict the age structure of a population at any time t given the initial age structure U (a, 0) and some conditions on birth and death rates, related to age. For example, the birth rate might be higher for times when the average age is lower and conversely the death rate might be higher for older populations. Other structured models may include independant variables such as size, weight etc.; such models are known as physiologically structured models. Other common PDE models in the life sciences include models for the biochemical reactions and motion of chemical species in the blood, cells or other media. Advection is most commonly used to model transport of particles, chemicals or organisms, via bulk motion of the supporting medium e.g. wind, water, blood etc. Diﬀusion models the random motions of particles etc. that cause these entities to disperse from high concentrations to lower concentrations. Sources and sinks depend upon the particular model. For population models the source term represents birth or growth rates, and a sink represents the death rate either by natural causes or by predation. For chemical problems, the source/sink term is called a reaction term and represents the rate at which a chemical species is created and/or consumed in chemical reactions, or lost by absorption (e.g. across a cell boundary). 3 Types of BCs and PDE classiﬁcation 3.1 Types of PDE Boundary Conditions As previously mentioned, to completely deﬁne a PDE model, the PDE equation itself is insuﬃcient. We also need to specify initial conditions that give the density at t = 0 for all points x in the domain Ω i.e. ε(x, 0) and boundary conditions specifying conditions on the density or ﬂux at the boundaries of the domain (δΩ) for all time i.e. ε(xb , t) or φ(xb , t) for xb ∈ δΩ. Typically the initial condition is simply some given function of space: ε(x, 0) = f (x), x∈Ω (9) There are three types of boundary conditions that often occur in physical problems: Dirichlet, Neumann and radiation boundary conditions. It is important to note that the boundary conditions for a given problem must be consistent with the speciﬁed PDE equation within the domain. For example, if we have a diﬀusion equation within the domain, it is invalid to specify an advection-like boundary condition and vice versa. 3.1.1 Dirichlet Boundary Conditions The ﬁrst type of boundary conditions, Dirichlet BCs, provide a function of time (g(t)) as the constraint on the density at a speciﬁed set of boundary points, xb ∈ (D ⊂ δΩ): ε(xb , t) = g(t), t>0 (10) 4 Suppose we are modelling heat ﬂow within the domain. The Dirichlet BC asserts that the temperature is constant along the speciﬁed set of boundary points xb . 3.1.2 Neumann Boundary Conditions The second type of BC is Neumann BCs, in which the ﬂux vector is speciﬁed by a given vector function (h(t)) along a subset of the domain boundary (N ): φ(xb , t) = h(t), t > 0, xb ∈ (N ⊂ δΩ) (11) The Neumann BC speciﬁes the inﬂux/outﬂux of the conserved quantity into/out of the domain from the surrounding environment. If h(t) = 0 then we say that the boundary is insulated ; there is no ﬂow across the boundary and the domain is isolated from the surrounding environment. 3.1.3 Radiation Boundary Conditions A third type of boundary condition of use for diﬀusive PDEs is the radiation or Robin BC: φ(xb , t) = β (ε(xb , t) − ψ(t)) , t>0 (12) In heat ﬂow, for example, this law expresses Newton’s law of cooling, which states that the heat ﬂux is proportional to the temperature diﬀerence between the domain boundary and the temperature of the surrounding environment, ψ(t). β is a heat-loss factor deﬁning the thermal coupling of the domain with the surrounding environment. In this case, one may wish to specify temperature proﬁle for the environment which evolves in time or which is a constant (ψ(t) =const.); in such situations we refer to the environment as a heat bath. 3.2 Classiﬁcation of PDEs We classify ODEs in terms of their order and whether they are linear or nonlinear. PDE models are more diﬃcult to classify, because of the greater variety of basic forms (or structures) the PDE may take. Not only are the order and linearity important, but also the PDE structure. It is the PDE structure that dictates what types of boundary and initial conditions can be imposed and ultimately what types of physical processes the PDE models. Let us consider a general second-order PDE in two independant variables. We can write such an equation in the form: Auxx + Buxt + Cutt + F (x, t, u, ux , ut ) = 0 (13) where A, B and C are constants. The classiﬁcation of this PDE is based upon the principal part of the equation, Lu ≡ Auxx + Buxt + Cutt . Here we have used x and t but this discussion applies equally well for two spatial variables also e.g. x and y. The classiﬁcation is based upon the sign of the quantity, D ≡ B 2 − 4AC which is called the discriminant. If D > 0, the equation is classiﬁed as hyperbolic; if D < 0 the equation is elliptic; and if D = 0 the equation is parabolic. This terminology comes from the classiﬁcation of plane curves; for example Ax2 + Ct2 = 1, where A, C > 0 and B = 0 (i.e. D < 0) graphs an ellipse in the xt-plane. Similarly, Ax2 − Ct2 = 1 graphs as a hyperbola. 5 Under this classiﬁcation, the diﬀusion equation is parabolic, Laplace’s equation is elliptic and the 1D wave equation (uxx = c2 utt ) is hyperbolic. Note that the advection equation does not have a classiﬁcation under this scheme; advection is a ﬁrst-order PDE. As it turns out, all parabolic equations are diﬀusion-like, all hyperbolic equations are wave-like, and all elliptic equations are static. 3.3 Characteristic coordinates and canonical forms Now we show that the principal part Lu can be simpliﬁed for all three types of PDEs by introducing a new set of independant variables. We seek a linear transformation of coordinates that simpliﬁes Lu: ξ = ax + bt, τ = cx + dt (14) where ξ and τ are new independant variables, and a, b, c, d are to be determined. So that we can transform back to the original coordinates, we require the transformation to be invertible i.e. ad − bc = 0. The dependent function u in the new variables will be denoted by U = U (ξ, τ ); that is, u(x, t) = U (ax + bt, cx + dt). Then using the chain rule for derivatives we can write the ﬁrst-order partial derivatives of u as: ux = Uξ ξx + Uτ τx = aUξ + cUτ ut = Uξ ξt + Uτ τt = bUξ + dUτ Applying the chain rule again, we obtain relations for the second-order partials: uxx = a2 Uξξ + 2acUξτ + c2 Uτ τ utt = b2 Uξξ + 2bdUξτ + d2 Uτ τ uxt = abUξξ + (ad + cb)Uξτ + cdUτ τ Substituting these quantities into the principal part and collecting like terms, we get Auxx + Buxt + Cutt = (Aa2 + Bab + Cb2 )Uξξ +(2acA + B(ad + bc) + 2Cbd)Uξτ +(Ac2 + Bcd + Cd2 )Uτ τ Now we can select a, b, c, d so that some of the partials in the new variables disappear. We have some ﬂexibility so we choose a = c = 1. Then the coeﬃcients of Uξξ and Uτ τ become quadratic expressions in b and d respectively and we can set these to zero to obtain values for b and d that eliminate Uξξ and Uτ τ from the equation e.g. A + Bb + Cb2 = 0 (15) has two possible solutions: √ −B ± D b= (16) 2C 6 To ensure the transformation is invertible, we need (d − b) = 0 so we can select the following values: √ √ −B + D −B − D b= and d = (17) 2C 2C 3.3.1 Hyperbolic case, D > 0 For the hyperbolic equation where D > 0, we can use the following transformation to characteristic coordinates: √ √ −B + D −B − D ξ =x+ tτ = x + t (18) 2C 2C to simplify the equation to its canonical form: Uξτ + G(ξ, τ, U, Uξ , Uτ ) = 0 (19) where only the mixed second-order partial derivative appears. This is a signiﬁcant simpli- ﬁcation over the original where all second-order derivatives occurred. Note that if C = 0 then these coordinates are not valid. We can however ﬁnd a diﬀerent set of characteristic coordinates by asserting that b = d = 1 instead. 3.3.2 Parabolic case, D = 0 For the parabolic case, Equations 17 give b = d which results in a non-invertible transfor- mation. After some trial and error, we can ﬁnd a suitable transformation that eliminates Uτ τ and Uξτ , namely: B ξ = x, τ =x− t (20) 2C i.e. a = c = 1 and b = 0, leaving d = B/2C. Therefore, the canonical form in the parabolic case becomes a diﬀusion-like equation: Uξξ + H(ξ, τ, U, Uξ , Uτ ) = 0 (21) 3.3.3 Elliptic case, D < 0 Equations 17 now yield values for b and d that are complex conjugates i.e. d = b. Selecting a = c = 1 again, we get a complex transformation: ξ = x + bt, τ = x + bt. (22) This is a little nasty but it can be avoided by making a second transformation to real variables α and β: 1 1 α = (ξ + τ ), β = (ξ − τ ). (23) 2 2i After a bit of routine algebra, the elliptic equation reduces to: Uαα + Uββ + K(α, β, U, Uα , Uβ ) = 0 (24) 7 where only the mixed partial is missing. Recognise that the combination of second-order partials is the Laplace Operator, so we have a nonhomogeneous Laplace equation for K = 0 or simply the homogeneous Laplace equation for K = 0. Generally transformation to characteristic coordinates is only of practical use for hyperbolic equations. 8 References 9